At this point, I believe I located a “finned” horizontal X-wing of 3's at r5c345 - r7c35. Since r4c5 can not be “3" if r5c4 (the “fin”) is “3", and it also cannot be “3" if the “finned” X-wing really turns out to be just an X-wing, in this case r4c5 must be set at 4. When r4c5 is set at 4, the puzzle collapses.

Questions: (1) Have I finally applied the “finned” X-wing rule (or at least perhaps a form of it) properly?; or (2) Did anyone use an alternate solution path to solve this puzzle?

I have no idea about 'finned x-wings', but from the position you posted I would note that the 4 in row 9 must be in box 7. It cannot be elsewhere in box 7, in particular it cannot be in r8c1. This leads to r2c1 being 4, from
here... well as you will see a lot of numbers fall immediately into place.

I'm sorry if this is not what you were hoping for, but I don't think it qualifies for some fancy name.

Let's see if I've got this right. There is a Finned X-Wing situation when:

1) An X-Wing would exist were it not for a third occurrence of the number in one row/column.

And

2) One of the wing's corners must be in a box with at least three occurrences of the number.

I assume it's that third occurrence which is called the "Fin", is that correct?

Does the Fin have to be in one of the boxes occupied by the wing's corners? For example, if the X-Wing were in boxes 4 and 5, could the Fin be in box 6?

Once the Finned Wing is recognized, it appears that we do some DIC's or something akin to them. Is there something in particular that should be looked for or some particular cell that should be a starting point? In other words, can this be reduced to some sort of rule, as in, once the pattern is observed, the next step must be such-and-such? Or is it a more open-ended thing?

Let's see if I've got this right. There is a Finned X-Wing situation when:
1) An X-Wing would exist were it not for a third occurrence of the number in one row/column.
And
2) One of the wing's corners must be in a box with at least three occurrences of the number.

I assume it's that third occurrence which is called the "Fin", is that correct?

Well, it's mostly right, Marty. But the "fin" doesn't have to be just one cell ... it can be two cells long. The logic is substantially similar in either case.

C. Same logic as "A" above, but since there's no complication at r9c3, once we eliminate the "x" at r5c7 there's an X-Wing on columns that also allows us to eliminate the fin itself, at r6c8.

D. Same logic as "A" above, but since there's no complication at r9c3, once we eliminate "x" at r4c7 and r5c7 there's an X-Wing on columns that also allows us to eliminate the fin itself, at r6c8.

E. Same logic as "B" above, but since there's no complication at r9c3, once we eliminate "x" at r4c7 and r5c7, there's an X-Wing on columns that also allows us to eliminate both cells' worth of "fin", at r6c8 & r6c9.

Many other variations are possible, but I think these examples give you the flavor of the general argument.

Marty R wrote:

Does the Fin have to be in one of the boxes occupied by the wing's corners? For example, if the X-Wing were in boxes 4 and 5, could the Fin be in box 6?

Yes, the "fin" must be in the same 3x3 box as one of the "corners" on the X-Wing. And it has to be in the same row (X-Wing on rows) or column (X-Wing on columns) as one leg of the X-Wing itself.

So the answer to your second question is no -- if the X-Wing lies in boxes 4 and 5, the "fin" must lie either in box 4 or in box 5; it cannot lie in box 6.

Marty R wrote:

... can this be reduced to some sort of rule, as in, once the pattern is observed, the next step must be such-and-such? Or is it a more open-ended thing?

Myth Jellies has reduced it to a general rule -- you can read about that in another forum.

Personally I find this rule to be of little practical utility. It's helpful if one is writing a computer program, but it's a little too much for my poor little brain to latch onto all at once. So in practice I tend to look for the general form of the pattern (an "almost X-Wing") and then go through the reasoning from first principles. But that's just me. dcb

As always, David, thank you very much for taking the time to explain things. I've printed out the reply, as I generally have difficulty following reasoning on a monitor, but can do so much better on paper.

Update: it's now a little over an hour since I posted the above and I've read through it once.

1) In all five examples, A-B-C-D-E, the chain of reasoning started with a finned cell, not one of the wing corners. Could that be one of the general "rules" I was looking for?

2) When this finned pattern exists, is it generally successful (eliminates at least one candidate) or does it often result in no action being taken?

Here the "*" in r9c2 is the external complication, the hash marks are the "fin", and the two asterisks at r4c8 & r5c8 represent possible eliminations in box 6.

Since there are six cells in box 6 lying outside of the "finned X-Wing" pattern, and two of those are ripe for exclusion because of the pattern, it looks like this version will be successful one-third of the time.

2. The general case, where there is no external complication, looks something like this.

Here we are guaranteed that we can either make the "dual exclusion" (first eliminate the target digit at r4c8 and at r5c8, followed by the "fin") or else we just have a regular X-Wing on columns (in which case the "fin" can be eliminated). So that's a 100% chance.

I'm not sure how much more likely case 1 is than case 2, but it appears that the overall probability of getting something useful out of the "finned" pattern is somewhere between one-third and 100% -- a 50% chance is probably a good estimate. dcb